In recent years, McLaughlin and Miller further presented a \(\bar{\partial }\)  steepest descent method which combine steepest descent with \(\bar{\partial }\) -problem rather than the asymptotic analysis
of singular integrals on contours to analyze asymptotic of orthogonal polynomials with non-analytical weights [1]}, [2]}.
When it is applied to integrable systems, the \(\bar{\partial }\)  steepest descent method also has displayed some advantages, such as avoiding
delicate estimates involving \(L^p\)  estimates of Cauchy projection operators, and leading the non-analyticity in the RHP reductions
to a \(\bar{\partial }\) -problem in some sectors of the complex plane which can be solved by being recast into an integral equation and by
using Neumann series. Dieng and McLaughin use it to study the defocusing NLS equation under essentially minimal regularity assumptions on finite mass
initial data [3]}; Cussagna and Jenkins study the defocusing NLS equation with finite density initial data [4]};
This \(\bar{\partial }\)  steepest descent method also was successfully
applied to prove asymptotic stability of N-soliton solutions to focusing NLS equation [5]};
Jenkins et.al studied soliton resolution for the derivative nonlinear NLS equation for generic initial data in a weighted Sobolev space [6]}. Their work
provided the soliton resolution property for derivative NLS equation, which decomposes the solution into the sum of a finite number of separated solitons and a
radiative parts when \(t\rightarrow \infty \) . And the dispersive part contains two components, one coming from the continuous spectrum and another from the interaction of the discrete and continuous spectrum.
